A categories and deformations

Mondays 5:10–6:25pm, 891 Evans, CCN 15536

Topic

We will explore the deformation theory of A algebras and categories, with special attention to the commutative case and the role of
Kontsevich formality. The role of curved algebras and the resulting problems may also be addressed, time permitting.

References will be updated.

Tentative schedule of lectures
Date
Topic
Speaker
References
9/10
Introduction and outline of the seminar
Teleman

9/17
Rational Homotopy I: DGAs
Teleman
2, 3, 4
9/24
Rational Homotopy II: DGLAs
Kiran
1, 3
10/1 Kuranishi construction of a germ of deformation spaces
Teleman
5
10/8
Formality of Kahler manifolds
Julian
4
10/15 Koszul duality for algebras Chris Kuo 6
10/22
Operads and examples: A∞, L, Comm, En
Yixuan 10
10/26
Operads and examples II: A∞, L, Comm, En
Yixuan 10. Note: Room 740 at 12pm
10/29 A algebras and categories
Meredith 7, 8
11/5
Bar complex, Hochschild complex, loop spaces and string topology?
Kiran
3 and more
11/26
Bar complex, Hochschild complex, wrapped up
Kiran

11/30
HKR theorem and formality statements
Teleman
12. 12:10–1:30pm, 891
12/3
Kontsevich Formality
Daniel
11, 12, 13
12/5
Deligne conjecture and Formality of E2 Non-formality
German
14; 12:10–1:30pm, 891
Next term
Curved algebras and their deformations

9


References

(Berkeley Library and sometimes source links)

  1. Quillen, Rational Homotopy Theory, Annals of Math 1969
  2. Sullivan, Infinitesimal computations in Topology, IHES Publ. Math. 1977
  3. Felix, Halperin, Thomas: Rational Homotopy. Springer GTM
  4. Deligne, Griffiths, Morgan, Sullivan: Real homotopy theory of Kaehler manifolds. Invent Math 1975
  5. Mimram, Koszul duality for quadratic algebras
  6. K. Kodaira: Complex manifolds and deformations of complex structures. Springer, Grundlehren
  7. B. Keller, Introduction to A-infinity algebras and modules
  8. B. Keller, On differential graded categories
  9. Keller, Lowen: On the (non)vanishing of some derived categories of curved dg algebras.
  10. Ginzburg and Kapranov, Koszul duality for operads
  11. Tamarkin, Formality of the chain operad of little disks
  12. Dolgushev, Tamarkin, Tsygan: Formality theorems for Hochschild complexes and their applications. Lett. Math. Phys 66 (2003)
  13. Lambrechts and Volic, Formality of the N-disk operad
  14. Dolgushev, Tamarkin, Tsygan: Proof of the Swiss Cheese version of Deligne's conjecture. Int. Math. Res. Not 2011